112 
Miscellanea 
Hence we deduce : 
l-''-i2^(l-fi')(l-M2') 
2i =0-1- 1 — :r-27i — r¥rjT—Y, 
= ^ (iii), 
Ml ^ Wj l-ri2^(l-/i.2-) Hi.2 Ml^n2 /J^.N 
<T2 l-r,.,^(l-/x,-'i)(l-M2') (^-i l-'-i2''(l-Mi-)(l-M2''') 
_ »«i /^2'''n2 1 -ri.>'^(l -^ i'-) , , 
(TO " <r, 1 - r,^ (1 - (1 - (U2^) ^ <T2 1 - ry^ (1 - Ml^) (1- M2^) ^ ^' 
These formulae will be fuuud useful (especially in deducing from Ki-,) in records in which 
there has been independent selection of two related individuals, without regard to their 
relationship. 
VII. On certain points concerning the Probable Error of the 
Standard Deviation"^'. 
By KAYMOND PEARL. 
The purpose of this paper is to discuss two problems of considerable practical importance in 
all biometrical investigations. These problems presented themselves in acute form in some 
studies of fecundity which the writer has at present under way. It was decided to be necessary 
to get definite answers to them before going farther with the work mentioned. In the belief 
that the matter is of general interest to workers in biometry it is presented here. The two 
points may be stated as follows : 
I. It has been shown t that if in any frequencj' distribution o-^i^ be the standard deviation 
for errors in the qth moment coefficient /i,^, taken about the mean, and o- be the standard 
deviation of the distribution, then 
where ii denotes the number in the sample. In this exjjression put q — and then, since 
fii=0, we have at once 
Probable error of /X2 = "67449 sj ^"^'^^ ^ 
Further since o- = a//xo, we have 
p.E. of 0-= -67449 ^'^J^^y?* (i). 
This is the true value of the probable error designated, whatever be the type of the frequency 
curves. But, for the normal curve, since there \i^ = 'i\x..^, (i) reduces at once to 
p.E. of 0-= -67449 
or as it is usually written =-67449-= (ii). 
* Papers from the Biological Laboratory of the Maine Agricultural Experiment Station, No. 1. 
t Biometrihi, Vol. ii. p. 276. 
